Abstract
We define the property “ E-cylindrical,” which relates to a subset of R m certain directed cylinders. We investigate some of the consequences of this definition, showing, for example, that polyhedral convex sets and smooth, rotund convex bodies are E-cylindrical. Suppose X is a finite set, F is the set of all real-valued functions on X, f ϵ F, and K ⊂ F is closed, convex, and E-cylindrical. For 1 < p < ∞, let f p be the best l p-approximation to f by elements of K. We show that lim p → ∞ f p exists. We give an example to show that { f p } may fail to converge if X is countably infinite. We discuss the relationship between discrete ( l p ) and continuous ( L p ) approximation.
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